Centrifugal Pump (TL)

Pressure source based on the centrifugal action of a rotating impeller

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Description

The Centrifugal Pump (TL) block models rotational conversion of energy from the shaft to the fluid in a thermal liquid network. The pressure differential and mechanical torque are modeled as a function of the pump head and brake power, which depend on pump capacity and are determined by linear interpolation of tabulated data. All formulations are based on the pump affinity laws, which scale pump performance to the ratio of current to reference values of the pump angular velocity and the fluid density. The differences in head due to fluid velocity and elevation change are not modeled.

Centrifugal Pump (TL) Block Schematic

Under nominal operating conditions, the fluid inlet is at port A and the fluid outlet is at port B. While the block supports reversed flows, flow from B to A is outside of the pump normal operating conditions. The pump mechanical reference associated with the pump casing is at port C and the shaft torque and rotational velocity are transferred at port R.

Analytical Parameterization: Capacity, Head, and Brake Power

The pressure gain over the pump is calculated as a function of the pump affinity laws and the reference pressure differential:

$$p_B-p_A=\Delta p_{ref}{\left(\frac{\omega }{{\omega }_{ref}}\right)}^2{\left(\frac{D}{D_{ref}}\right)}^2,$$

where:

• Δp_ref is the reference pressure gain, which is determined from a quadratic fit of the pump pressure differential between the Maximum head at zero capacity, the Nominal head, and the Maximum capacity at zero head.

• ω is the shaft angular velocity, ω_R – ω_C.

• ω_ref is the Reference shaft speed.

• $$\frac{D}{D_{ref}}$$ is the Impeller diameter scale factor, which can be modified from the default value of 1 if your reference and system impeller diameters differ. This block does not reflect changes in pump efficiency due to pump size.

• ρ is the network fluid density.

The shaft torque is:

$$\tau =W_{brake,ref}\frac{{\omega }^2}{{\omega }_{ref}^3}{\left(\frac{D}{D_{ref}}\right)}^5.$$

The reference brake power, W_brake,ref is determined from a linear fit between the Nominal brake power and the Brake power at zero capacity.

The reference capacity is calculated as:

$$q_{ref}=\frac{\dot{m}}{\rho }\frac{{\omega }_{ref}}{\omega }{\left(\frac{D_{ref}}{D}\right)}^3.$$

1-D Tabulated Data Parameterization: Head and Brake Power as a Function of Capacity

You can model pump performance as a 1-D function of capacity, the volumetric flow rate through the pump. The pressure gain over the pump is based on a reference shaft speed and is a function of the Reference head vector, ΔH_ref, evaluated at a reference capacity, Q_ref:

$$\Delta p=\rho g\Delta H_{ref}(Q_{ref})\frac{{\omega }^2}{{\omega }_{ref}^2}{\left(\frac{D}{D_{ref}}\right)}^2,$$

where:

• ω is the shaft angular velocity.

• ρ is the fluid density.

• g is the gravitational constant.

This is derived from the affinity law that relates head and angular velocity:

$$\frac{\Delta H_{ref}}{\Delta H}=\frac{{\omega }_{ref}^2}{{\omega }^2}{\left(\frac{D}{D_{ref}}\right)}^2,$$

where ΔH is the head.

The shaft torque is based on the Reference brake power vector, P_ref, which is a function of the reference capacity, Q_ref:

$$T=P_{ref}(Q_{ref})\frac{{\omega }^2}{{\omega }_{ref}^3}\frac{\rho }{{\rho }_{ref}}{\left(\frac{D}{D_{ref}}\right)}^5,$$

where ρ_ref is the fluid Reference density.

This is derived from the affinity law that relates brake power and angular velocity:

$$\frac{P_{ref}}{P}=\frac{{\omega }_{ref}^3}{{\omega }^3}\frac{{\rho }_{ref}}{\rho }{\left(\frac{D}{D_{ref}}\right)}^5.$$

The reference capacity is defined as:

$$Q_{ref}=\frac{\dot{m}}{\rho }\frac{{\omega }_{ref}}{\omega }{\left(\frac{D}{D_{ref}}\right)}^3,$$

where $$\dot{m}$$ is the mass flow rate at the pump inlet.

If the simulation is beyond the bounds of the provided table, the pump head is extrapolated linearly and brake power is extrapolated to the nearest point.

2-D Tabulated Data Parameterization: Head and Brake Power as a Function of Capacity and Shaft Speed

You can model pump performance as a 2-D function of capacity and shaft angular velocity. The pressure gain over the pump is a function of the Head table, H(Q,w), ΔH_ref, which is a function of the reference capacity, Q_ref, and the shaft speed, ω:

$$\Delta p=\rho g\Delta H_{ref}(Q_{ref},\omega ){\left(\frac{D}{D_{ref}}\right)}^2.$$

The shaft torque is calculated as a function of the Brake power table, Wb(q,w), P_ref, which is a function of the reference capacity, Q_ref, and the shaft speed, ω:

$$T=\frac{P_{ref}(Q_{ref},\omega )}{{\omega }_{ref}}\frac{\rho }{{\rho }_{ref}}{\left(\frac{D}{D_{ref}}\right)}^5.$$

The reference capacity is calculated as:

$$Q_{ref}=\frac{\dot{m}}{\rho }{\left(\frac{D_{ref}}{D}\right)}^3.$$

If the simulation is beyond the bounds of the provided table, the pump head is extrapolated linearly and brake power is extrapolated to the nearest point.

Mechanical Orientation

The pump generates power when the shaft at port R rotates in the same direction as the Mechanical orientation setting. Setting this parameter to Positive angular velocity of port R relative to port C corresponds to normal pump operation means that fluid flows from A to B when R rotates in the positive convention relative to port C. When the shaft rotates counter to the Mechanical orientation setting, torque is generated, but it may not be physically accurate.

Energy Balance

Mechanical work done by the pump is associated with an energy exchange. The governing energy balance equation is:

$${\varphi }_A+{\varphi }_B+P_{mech}=0,$$

where:

• Φ_A is the energy flow rate at port A.

• Φ_B is the energy flow rate at port B.

• P_mech is the mechanical power generated due to torque, T, and the pump angular velocity, ω: $$P_{mech}=T\omega .$$

The pump hydraulic power is a function of the pressure difference between pump ports:

$$P_{hydro}=\Delta p\frac{\dot{m}}{\rho }.$$

Ports

Conserving

A — Pump inlet

thermal liquid

Fluid inlet port.

B — Pump outlet

thermal liquid

Fluid outlet port.

C — Pump casing

mechanical rotational

Casing angular velocity and torque.

R — Impeller shaft

mechanical rotational

Shaft angular velocity and torque.

Parameters

Pump Parameterization — Pump performance data

Capacity, head, and brake power at reference shaft speed (default) | 1D tabulated data - head and brake power vs. capacity | 2D tabulated data - head and brake power vs. capacity and angular velocity

Parameterization of the pump head and brake power.

• Capacity, head, and brake power at reference shaft speed: Model pump pressure gain and shaft torque with an analytical formula.

• 1D tabulated data - head and brake power vs. capacity at reference shaft speed: Model head and brake power from tabulated data of head and brake power at a given capacity.

• 2D tabulated data - head and brake power vs. capacity and shaft speed: Model head and brake power from tabulated data of head and brake power at a given capacity and shaft speed.

Nominal capacity — Pump volumetric flow rate

45 lpm (default) | positive scalar

Nominal pump volumetric flow rate at a reference shaft angular velocity.

Dependencies

To enable this parameter, set Pump parameterization to Capacity, head, and brake power at reference shaft speed.

Nominal head — Pump head

40 m (default) | positive scalar

Nominal pump pressure differential, normalized by gravity and the fluid density, at a reference shaft angular velocity.

Dependencies

To enable this parameter, set Pump parameterization to Capacity, head, and brake power at reference shaft speed.

Nominal brake power — Pump power

0.85 kW (default) | positive scalar

Nominal mechanical shaft power at a reference angular velocity.

Dependencies

To enable this parameter, set Pump parameterization to Capacity, head, and brake power at reference shaft speed.

Maximum head at zero capacity — Maximum pump head with no flow

60 m (default) | positive scalar

Maximum pump head with no flow at a reference angular velocity. This parameter is used to determine the reference pressure differential over the pump in the analytical parameterization by providing the roots of the quadratic equation for pressure in conjunction with the Nominal capacity, the Nominal head, and the Maximum capacity at zero head parameters.

Dependencies

To enable this parameter, set Pump parameterization to Capacity, head, and brake power at reference shaft speed.

Brake power at zero capacity — Maximum pump power without flow

0.5 kW (default) | positive scalar

Maximum pump power without flow at a reference angular velocity. This parameter is used to determine the reference pump torque in the analytical parameterization by providing the roots of the equation for reference friction torque, in conjunction with the Nominal brake power.

Dependencies

To enable this parameter, set Pump parameterization to Capacity, head, and brake power at reference shaft speed.

Maximum capacity at zero head — Maximum flow rate with zero head

80 lpm (default) | positive scalar

Maximum fluid load with zero head at a reference angular velocity. This parameter is used to determine the reference pressure differential over the pump in the analytical parameterization by providing the roots of the quadratic equation for pressure in conjunction with the Nominal capacity, the Nominal head, and the Maximum head at zero capacity parameters.

Dependencies

To enable this parameter, set Pump parameterization to Capacity, head, and brake power at reference shaft speed.

Reference shaft speed — Reference shaft angular velocity for affinity law calculations

1770 rpm | 3500 rpm | positive scalar

Reference angular velocity for affinity law calculations. The default value depends on the Pump parameterization setting.

Dependencies

To enable this parameter, set Pump parameterization to either:

• Capacity, head, and brake power at reference shaft speed.

• 1D tabulated data - head and brake power vs. capacity at reference shaft speed.

Reference capacity vector — Volumetric flow rate

[16.67, 30, 40, 53.33, 66.67, 83.33] lpm (default) | 1-by-n vector

Vector of volumetric flow rates for the tabular parameterization of pump head or brake power. This parameter corresponds one-to-one with the Reference head vector and Reference brake power vector parameters. The vector elements are listed in ascending order and must be greater than 0.

Dependencies

To enable this parameter, set Pump parameterization to 1D tabulated data - head and brake power vs. capacity at reference shaft speed.

Reference head vector — Vector of pump head values for tabulated data parameterization

[50, 47.5, 45, 40, 35, 25] m (default) | 1-by-n vector

Vector of pump head values for the 1-D tabular parameterization of pump head and brake power. This parameter corresponds one-to-one with the Reference capacity vector parameter. The vector elements must be listed in descending order and must be greater than or equal to 0.

Dependencies

To enable this parameter, set Pump parameterization to 1D tabulated data - head and brake power vs. capacity at reference shaft speed.

Reference brake power vector — Vector of pump brake power values for tabulated data parameterization

[.6, .75, .85, .93, .98, .99] kW (default) | 1-by-n vector

Vector of pump brake power values for the 1-D tabular parameterization of pump head and brake power. This parameter corresponds to the Reference capacity vector parameter. Vector elements must be greater than 0.

Dependencies

To enable this parameter, set Pump parameterization to 1D tabulated data - head and brake power vs. capacity at reference shaft speed.

Capacity vector, q — Vector of volumetric displacement for tabulated data parameterization

[16.67, 23.33, 30, 40, 53.33, 60] lpm (default) | 1-by-n vector

Vector of volumetric flow rates for the tabular parameterization of pump head. This vector forms an independent axis with the Shaft speed vector, w parameter for the 2-D Head table, H (q,w) and Brake power table, Wb(q,w) parameters. The vector elements must be listed in ascending order and must be greater than or equal to 0.

Dependencies

To enable this parameter, set Pump parameterization to 2D tabulated data - head and brake power vs. capacity and shaft speed.

Shaft speed vector, w — Vector of angular velocities for tabulated data parameterization

[2450, 2800, 3150, 3500] rpm (default) | 1-by-n vector

Vector of shaft angular velocity values for the tabular parameterization of pump head. This vector forms an independent axis with the Capacity vector, q parameter for the 2-D Head table, H (q,w) and Brake power table, Wb(q,w) parameters. The vector elements must be listed in ascending order and must be greater than 0.

Dependencies

To enable this parameter, set Pump parameterization to 2D tabulated data - head and brake power vs. capacity and shaft speed.

Head table, H(q,w) — Pump head for tabulated data parameterization

[25, 32.7, 42.5, 50; 24, 31.7, 41, 49; 22.7, 30.5, 39, 47.5; 20, 28, 36, 45; 15.7, 24, 32, 40; 12.5, 21, 29.5, 37.5] m (default) | M-by-N matrix

M-by-N matrix of pump head values at the specified volumetric flow rate and angular velocity. All table elements must be greater than or equal to 0. Linear interpolation is employed between table elements. M and N are the sizes of the corresponding vectors:

• M is the number of vector elements in the Capacity vector, q parameter. All columns of the array must be in strictly descending order. For example, H(1,1) must be the largest value in the column, and H(M,1) must be the smallest.

• N is the number of vector elements in the Shaft speed vector, w parameter. All rows must be in strictly ascending order.

Dependencies

To enable this parameter, set Pump parameterization to 2D tabulated data - head and brake power vs. capacity and shaft speed.

Brake power table, Wb(q,w) — Pump brake power for tabulated data parameterization

[.28, .38, .5, .6; .31, .41, .55, .68; .33, .44, .6, .75; .36, .48, .65, .83; .38, .53, .72, .93; .39, .55, .75, .97] kW (default) | M-by-N matrix

M-by-N matrix of pump brake power values at the specified volumetric flow rate and angular velocity. All values must be greater than 0. Linear interpolation is employed between table elements. M and N are the sizes of the corresponding vectors:

• M is the number of vector elements in the Capacity vector, q parameter.

• N is the number of vector elements in the Shaft speed vector, w parameter. All rows must be in strictly ascending order.

Dependencies

To enable this parameter, set Pump parameterization to 2D tabulated data - head and brake power vs. capacity and shaft speed.

Impeller diameter scale factor — Ratio of model diameter to reference diameter for affinity law calculations

1 (default) | positive scalar

Ratio of model diameter to reference diameter for affinity law calculations. Modify this value if there is a difference between your reference and system impeller diameters, such as when testing pump scaling. For system pumps smaller than the reference pump, use a value less than 1. For system pumps larger than the reference pump, use a value grater than 1. This block does not reflect changes in pump efficiency due to pump size.

Mechanical orientation — Shaft normal rotational direction

Positive angular velocity of port R relative to port C corresponds to normal pump operation (default) | Negative angular velocity of port R relative to port C corresponds to normal pump operation

Shaft rotational direction for flow from port A to B. This parameter identifies the positive and reverse shaft rotation of R with respect to C.

Reference density — Fluid density for which the pump performance data is reported

998 kg/m^3 (default) | positive scalar with units of mass/volume

Fluid density specified in reference or data sheet performance. This parameter is used to scale pump performance between different fluids.

Cross-sectional area at ports A and B — Area normal to the direction of flow at the ports

0.01 m^2 (default) | positive scalar with units of area

Area at ports A and B.

Model Examples

Aircraft Fuel Supply System with Three Tanks

The fuel supply system represented in the example consists of three tanks and an engine. The tanks are connected by the fuel lines and the pressure drop in the fuel lines depends on the bank angle of the aircraft or the relative elevation of the ends of the fuel lines. The central tank bottom is elevated by 36 in with respect to the reference plane, which is drawn horizontally through the entry point of the engine. The side tank bottoms are elevated by 4.2 in each at zero bank angle.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.